The Cobblers had taken an early away lead but goals from Dennis Lawrence and Andy Robinson helped keep Swansea's play-off hopes alive.
Brad Johnson gave Northampton a surprise lead when he drove home a 30-yard free kick on six minutes, after Ian Craney had pulled down Ryan Gilligan.
The home side had a penalty shout turned down on 23 minutes when Pawel Abbott went down in the box under a challenge from Northampton's Mark Hughes.
Andy Robinson almost grabbed The Swans an equaliser five minutes before half time but his right-foot shot was pushed against the crossbar by goalkeeper Mark Bunn.
Swansea equalised on the stroke of half time when Dennis Lawrence collected Alan Tate's speculative drive on the edge of the box and passed the ball into the bottom corner.
Northampton were caught out 24 seconds into the second half when Tom Butler's deep cross found Andy Robinson on his own at the back post and the midfielder made no mistake from four yards.
Johnson nearly equalised for Northampton three minutes later but his fierce left-foot volley came back off the Swansea crossbar.
Butler and Leon Britton began to cause Northampton problems with their direct running from midfield. Butler was unlucky not to score after a fine run on 50 minutes but his shot flew high over the Northampton goal.
Swansea keeper Willy Gueret nearly gifted Northampton a goal when he flapped at a cross on 60 minutes but Hughes' shot was cleared off the line by defender Izzy Iriekpen.
Ian Craney and Iriekpen both went close for The Swans as the half wore on but could not beat keeper Bunn.
Northampton pressed for an equaliser in the closing minutes but Swansea held on for a vital victory.